

Brahmagupta’s formula 51

If xis a real number, then the bracket symbols x,

x, and {x} are used to denote the floor, ceiling, and

fractional part values, respectively, of x.

Square brackets [ ] and parentheses ( ) are placed at

the end points of an

INTERVAL

on the real number line

to indicate whether or not the end points of that inter-

val are to be included.

See also

EXPANDING BRACKETS

;

FLOOR

/

CEILING

/

FRACTIONAL PART FUNCTIONS

;

ORDER OF OPERATION

.

Brahmagupta (ca. 598–665) Indian Arithmetic, Geo-

metry, Astronomy Born in Ujjain, India, scholar Brah-

magupta is recognized as one of the important

mathematicians of the seventh century. His famous 628

text Brahmasphutasiddhanta (The opening of the uni-

verse) on the topic of astronomy includes such notable

mathematical results as his famous formula for the

AREA

of a cyclic

QUADRILATERAL

, the integer solution to

certain algebraic equations, and methods of solution to

simultaneous equations. This work is also historically

significant as the first documented systematic use of

ZERO

and negative quantities as valid numbers in

ARITHMETIC

.

Brahmagupta was head of the astronomical obser-

vatory at Ujjain, the foremost mathematical center of

ancient India, and took an avid interest in the develop-

ment of astronomical observation and calculation. The

first 10 of the 25 chapters of Brahmasphutasiddhanta

pertain solely to astronomy, discussing the longitude of

the planets, lunar and solar eclipses, and the timing of

planet alignments. Although rich in mathematical com-

putation and technique, it is the remainder of the work

that offers an insight into Brahmagupta’s far-reaching

understanding of mathematics on an abstract level.

Brahmagupta goes on to describe the decimal

PLACE

-

VALUE SYSTEM

used in India at his time for representing

numerals and the methods for doing arithmetic in this

system. (For instance, he outlines a method of “long

multiplication” essentially equivalent to the approach we

use today.) Brahmagupta permits zero as a valid number

in all of his computations, and in fact gives it the explicit

status of a number by defining it as the result of sub-

tracting a quantity from itself. (Until then, zero acted as

nothing more than a placeholder to distinguish 203 from

23, for instance.) He also explains the arithmetical prop-

erties of zero—that adding zero to a number leaves that

number unchanged and multiplying any number by zero

produces zero, for instance. Brahmagupta also detailed

the arithmetic of negative numbers (which he called

“debt”) and suggested, for the first time, that they may

indeed be valid solutions to certain problems.

Brahmagupta next explores problems in

ALGEBRA

.

He develops some basic algebraic notations and then

presents a series of methods for solving a variety of lin-

ear and quadratic equations. For instance, he devised

an ingenious technique for finding integer solutions to

equations of the form ax2+ c= y2. (For example, Brah-

magupta correctly asserted that x= 226,153,980 and y

= 1,766,319,049 are the smallest positive integer solu-

tions to 61x2+ 1 = y2.) Brahmagupta also presents the

famous

SUMS OF POWERS

formulae:

as well as algorithms for computing square roots.

Unfortunately, as was the practice of writing at the

time, Brahmagupta never gave any word of explana-

tion as to how his solutions or formulae were found.

No proofs were ever offered.

In the final sections of Brahmasphutasiddhanta,

Brahmagupta presents his famous formula for the area

of a cyclic quadrilateral solely in terms of the lengths

of its sides. Curiously, Brahmagupta does not state

that the formula is true only for quadrilaterals

inscribed in a

CIRCLE

.

In a second work, Khandakhadyaka, written in

665, Brahmagupta discusses further topics in astron-

omy. Of particular interest to mathematicians, Brah-

magupta presents here an ingenious method for

computing values of sines.

Brahmagupta’s methods and discoveries were

extremely influential. Virtually every text that dis-

cusses Indian astronomy describes or uses some aspect

of his work.

See also B

RAHMAGUPTA

’

S FORMULA

.

Brahmagupta’s formula Seventh-century Indian math-

ematician and astronomer B

RAHMAGUPTA

derived a for-

mula for the

AREA

of a

QUADRILATERAL

inscribed in a